What is the derivative of sin(sin(cos(x)sin(x)))?

Sep 27, 2016

$= \cos \left(\sin \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \left(\cos \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \cos \left(2 x\right)$

Explanation:

I assume that x is in radian measure so that

$\left(\sin x\right) ' = \cos x \mathmr{and} \left(\cos x\right) ' = - \sin x$.

Applying chain rule,

(sin(sin(cos(x)sin(x)))'

=cos(sin(cos(x)sin(x)))(sin((cos(x)sin(x))'

$= \cos \left(\sin \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \left(\cos \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \left(\cos \left(x\right) \left(\sin \left(x\right)\right) ' + \left(\cos \left(x\right)\right) ' \sin \left(x\right)\right)$

$= \cos \left(\sin \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \left(\cos \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \left({\cos}^{2} \left(x\right) - {\sin}^{2} \left(x\right)\right)$

$= \cos \left(\sin \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \left(\cos \left(\cos \left(x\right) \sin \left(x\right)\right)\right) \cos \left(2 x\right)$